Question

The dosage of medicine that a person is prescribed is often determined by the person's weight. The following computer output and residual plot for the exponential model were provided. What is the predicted dosage for someone who weighs 208 pounds? 1) 10 mg 2) 13 mg 3) 182 mg 4) 353 mg

Answer 1

The predicted dosage for someone who weighs 208 pounds, based on the exponential model provided in the computer output and residual plot, is approximately 182 mg. Therefore, the correct option is 3) 182 mg.

Step-by-step explanation:

The computer output and residual plot suggest that an exponential model is being used to predict the dosage based on weight. The exponential model is typically represented as:

`\[ \text{Dosage} = a \cdot e^{b \cdot \text{Weight}} \]`

where:

- Dosage is the predicted dosage,

- Weight is the weight of the person,

- a is the initial dosage, and

- b is a coefficient determined by the model.

To find the predicted dosage for someone who weighs 208 pounds, substitute Weight= 208 into the exponential model using the coefficients obtained from the computer output. The resulting predicted dosage is approximately 182 mg.

Therefore, the correct option is 3) 182 mg, as this represents the predicted dosage for a person weighing 208 pounds based on the provided exponential model.

Answer 2

the model appears to be a good fit with a strong
`\( R^2 \)`, and the residual plot does not show a pattern, which are indicative of a well-fitting model. However, without more information, we cannot confirm whether the model compares the log of weight to the log of dosage or if the "value of 2" refers to some aspect of the
`\( R^2 \)` prediction.

Based on the provided computer output and residual plot for the exponential model, we can make several observations to determine which statements apply:

1. The residual plot is patterned:

- A residual plot displays the residuals (the differences between observed and predicted values) on the vertical axis and the independent variable (in this case, weight) on the horizontal axis.

- A patterned residual plot would indicate that there is a systematic variation that the model has not captured. However, from the image provided, the residuals appear to be randomly scattered around the horizontal axis without an obvious pattern, so this statement does not apply.

2. The model is a good fit:

- Model fit can be assessed by several indicators, including the
`\( R^2 \)` value, the residual plot, p-values of the coefficients, etc.

- The provided output shows a high
`\( R^2 \)` value and the residuals seem randomly scattered, which generally indicates a good fit.

3. The
`\( R^2 \)` is considered strong:

- The
`\( R^2 \)` value (coefficient of determination) indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

- An
`\( R^2 \)` value of 99.45% (or adjusted
`\( R^2 \)` of 99.37%) is extremely high, suggesting that the model explains nearly all the variability in the response data around its mean. Thus, the
`\( R^2 \)` can be considered strong.

4. The value of 2 is 99.08%:

- It is unclear what is meant by "the value of 2" in this context. If this is referring to the
`\( R^2 \)` value or some aspect of the
`\( R^2 \)` prediction, then the statement is incorrect. The
`\( R^2 \)` value is 99.45% and the
`\( R^2 \)` (predicted) is 99.08%.

5. The transformed model compares the log of weight to the log of dosage:

- A transformed model might take the logarithm of variables to linearize relationships, especially in the context of exponential growth or decay.

- The residual plot is titled "Residuals vs. Weight (Response is Log(Dosage))," which indicates that the log of dosage is the response variable being modeled against the weight. This suggests that the model uses the log of dosage but does not specify that the log of weight is used. Therefore, without further information, we cannot conclude that the log of weight is compared to the log of dosage.

Based on the information from the output and the residual plot, the model appears to be a good fit with a strong
`\( R^2 \)`, and the residual plot does not show a pattern, which are indicative of a well-fitting model. However, without more information, we cannot confirm whether the model compares the log of weight to the log of dosage or if the "value of 2" refers to some aspect of the
`\( R^2 \)` prediction.